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Electrostatics
Electrostatics is a branch of that studies s at rest. Electrostatic phenomena arise from the s that electric charges exert on each other. Such forces are described by . Coulomb's law Coulomb's law states that: 'The magnitude of the electrostatic force of attraction or repulsion between two point charges is directly proportional to the product of the magnitudes of charges and inversely proportional to the square of the distance between them.' The force is along the straight line joining them. If the two charges have the same sign, the electrostatic force between them is repulsive; if they have different signs, the force between them is attractive. If r is the distance (in ) between two charges, then the force (in s) between two point charges q and Q (in s) is: : F = \frac{1}{4\pi \varepsilon_0}\frac{qQ}{r^2}= k_0\frac{qQ}{r^2}\, , where e0 is the , or permittivity of free space: : \varepsilon_0 \approx {10^{-9}\over 36\pi} \;\; \mathrm{C^2\ N^{-1}\ m^{-2}}\approx 8.854\ 187\ 817 \times 10^{-12} \;\; \mathrm{C^2\ N^{-1}\ m^{-2}}. The units of e0 are equivalently   2 4 kg-1m-3 or 2 −1m−2 or m−1. is: : k_0 = \frac{1}{4\pi\varepsilon_0}\approx 8.987\ 551\ 787 \times 10^9 \;\; \mathrm{N\ m^2\ C}^{-2}. A single has a charge of e'', and the has a charge of -''e, where, : e \approx 1.602\ 176\ 565 \times 10^{-19}\;\; \mathrm{C}. These s (e0, k0, e) are currently defined so that e0 and k0 are exactly defined, and e'' is a measured quantity. Electric field ''(lines with arrows) of a nearby positive charge (+) causes the mobile charges in conductive objects to separate due to . Negative charges (blue) are attracted and move to the surface of the object facing the external charge. Positive charges (red) are repelled and move to the surface facing away. These induced surface charges are exactly the right size and shape so their opposing electric field cancels the electric field of the external charge throughout the interior of the metal. Therefore, the electrostatic field everywhere inside a conductive object is zero, and the is constant.}} The , \vec{E} , in units of per or s per meter, is a that can be defined everywhere, except at the location of point charges (where it diverges to infinity). It is defined as the electrostatic force \vec{F}\, in newtons on a hypothetical small at the point due to , divided by the magnitude of the charge q\, in coulombs : \vec{E} = {\vec{F} \over q}\, are useful for visualizing the electric field. Field lines begin on positive charge and terminate on negative charge. They are parallel to the direction of the electric field at each point, and the density of these field lines is a measure of the magnitude of the electric field at any given point. Consider a collection of N particles of charge Q_i , located at points \vec r_i (called source points), the electric field at \vec r (called the field point) is: : \vec{E}(\vec r) =\frac{1}{4\pi \varepsilon _0}\sum_{i=1}^N \frac{\widehat R_i Q_i}{\left \|\vec R_i \right \|^2} , where \vec R_i = \vec r - \vec r_i , is the displacement vector from a source point \vec r_i to the field point \vec r , and \widehat R_i = \vec R_i / \left \|\vec R_i \right \| is a that indicates the direction of the field. For a single point charge at the origin, the magnitude of this electric field is E =k_eQ/ R^2, and points away from that charge is positive. The fact that the force (and hence the field) can be calculated by summing over all the contributions due to individual source particles is an example of the . The electric field produced by a distribution of charges is given by the volume \rho (\vec r) and can be obtained by converting this sum into a : : \vec{E}(\vec r)= \frac {1}{4 \pi \varepsilon_0} \iiint \frac {\vec r - \vec r \,'}{\left \| \vec r - \vec r \,' \right \|^3} \rho (\vec r \,')\operatorname{d}^3 r\,' Gauss' law states that "the total electric flux through any closed surface in free space of any shape drawn in an electric field is proportional to the total enclosed by the surface." Mathematically, Gauss's law takes the form of an integral equation: : \oint_S\vec{E} \cdot\mathrm{d}\vec{A} = \frac{1}{\varepsilon_0}\,Q_{enclosed}=\int_V{\rho\over\varepsilon_0}\cdot\operatorname{d}^3 r, where \operatorname{d}^3 r =\mathrm{d}x \ \mathrm{d}y \ \mathrm{d}z is a volume element. If the charge is distributed over a surface or along a line, replace \rho\mathrm{d}^3r by \sigma\mathrm{d}A or \lambda\mathrm{d}\ell . The allows Gauss's Law to be written in differential form: : \vec{\nabla}\cdot\vec{E} = {\rho\over\varepsilon_0}. where \vec{\nabla} \cdot is the . Poisson and Laplace equations The definition of electrostatic potential, combined with the differential form of Gauss's law (above), provides a relationship between the potential F and the charge density ?: : {\nabla}^2 \phi = - {\rho\over\varepsilon_0}. This relationship is a form of . In the absence of unpaired electric charge, the equation becomes : : {\nabla}^2 \phi = 0, Electrostatic approximation The validity of the electrostatic approximation rests on the assumption that the electric field is : : \vec{\nabla}\times\vec{E} = 0. From , this assumption implies the absence or near-absence of time-varying magnetic fields: : {\partial\vec{B}\over\partial t} = 0. In other words, electrostatics does not require the absence of magnetic fields or electric currents. Rather, if magnetic fields or electric currents do exist, they must not change with time, or in the worst-case, they must change with time only very slowly. In some problems, both electrostatics and may be required for accurate predictions, but the coupling between the two can still be ignored. Electrostatics and magnetostatics can both be seen as Galilean limits for electromagnetism. Electrostatic potential Because the electric field is , it is possible to express the electric field as the of a scalar function, \phi , called the (also known as the ). An electric field, E , points from regions of high electric potential to regions of low electric potential, expressed mathematically as : \vec{E} = -\vec{\nabla}\phi. The can be used to establish that the electrostatic potential is the amount of per unit charge required to move a charge from point a to point b with the following : : -\int_a^b{\vec{E}\cdot \mathrm{d}\vec \ell} = \phi (\vec b) -\phi(\vec a). From these equations, we see that the electric potential is constant in any region for which the electric field vanishes (such as occurs inside a conducting object). Electrostatic energy A single 's potential energy, U_\mathrm{E}^{\text{single}} , can be calculated from a of the work, q_n\vec E\cdot\mathrm d\vec\ell . We integrate from a point at infinity, and assume a collection of N particles of charge Q_n , are already situated at the points \vec r_i . This potential energy (in s) is: : U_\mathrm{E}^{\text{single}}=q\phi(\vec r)=\frac{q }{4\pi \varepsilon_0}\sum_{i=1}^N \frac{Q_i}{\left \|{\vec R_i} \right \|} where \vec {R_i} = \vec r - \vec r_i is the distance of each charge Q_i from the q , which situated at the point \vec r , and \phi(\vec r) is the electric potential that would be at \vec r if the were not present. If only two charges are present, the potential energy is k_eQ_1Q_2/r . The total due a collection of N'' charges is calculating by assembling these particles : : U_\mathrm{E}^{\text{total}} = \frac{1 }{4\pi \varepsilon _0}\sum_{j=1}^N Q_j \sum_{i=1}^{j-1} \frac{Q_i}{r_{ij}}= \frac{1}{2}\sum_{i=1}^N Q_i\phi_i , where the following sum from, ''j = 1 to N'', excludes ''i = j: : \phi_i = \frac{1}{4\pi \varepsilon _0}\sum_{j=1 (j\ne i)}^N \frac{Q_j}{r_{ij}}. This electric potential, \phi_i is what would be measured at \vec r_i if the charge Q_i were missing. This formula obviously excludes the (infinite) energy that would be required to assemble each point charge from a disperse cloud of charge. The sum over charges can be converted into an integral over charge density using the prescription \sum (\cdots) \rightarrow \int(\cdots)\rho\mathrm d^3r : : U_\mathrm{E}^{\text{total}} = \frac{1}{2} \int\rho(\vec{r})\phi(\vec{r}) \operatorname{d}^3 r = \frac{\varepsilon_0 }{2} \int \left|{\mathbf{E}}\right|^2 \operatorname{d}^3 r , This second expression for uses the fact that the electric field is the negative of the electric potential, as well as in a way that resembles . These two integrals for electric field energy seem to indicate two mutually exclusive formulas for electrostatic energy density, namely \frac{1}{2}\rho\phi and \frac{\varepsilon_0 }{2}E^2 ; they yield equal values for the total electrostatic energy only if both are integrated over all space. Electrostatic pressure On a , a surface charge will experience a force in the presence of an . This force is the average of the discontinuous electric field at the surface charge. This average in terms of the field just outside the surface amounts to: : P = \frac{ \varepsilon_0 }{2} E^2 , This pressure tends to draw the conductor into the field, regardless of the sign of the surface charge. References Category:Electricity